By: Amir Ronen, Department of CS Stanford University Presented By: Oren Mizrahi Matan Protter Issues on border of economics & computation, 2002

We will discuss the issue of revenue maximization, also known as optimal auction design. It is a subject of long and intensive research in microeconomics. We will look for an approximation.

[ n ] = { 0, 1, 2,.., n} W i = {1, 1 + ε, ε, …, 2, 2 + ε, …, h } : The possible types (valuations ) of each agent. Φ = A distribution over the type space. R m = The revenue of the auction m = The expected payment

An Auction: A pair of function (k,p) such that: K : W [n] is an allocation algorithm determining who wins the object (a zero – no winner). P : W R is a payment function determining how much the winner must pay.

C – Approximation: An Auction m is a C-approximation over Φ if for every valid auction v’,. If c=1, the auction is optimal. A Valid Auction: An auction the satisfies both: Individual Rationality (IR): The profit of a truth telling agent is always non – negative: p(w) ≤ w k(w). Incentive Compatibility (IC): Truth-telling is a dominant strategy for each agent.

An Algorithm with the following charecaristics: Input: One item to sell. A probability distribution over the type space. Constant C. Output: An auction. Restrictions: Auction is a C-approximation optimal auction. Both Algorithm and auction are polytime.

Suppose Alice wishes to sell a house to either Bob1 or Bob2, for prices in the range [0,100]. Let’s look at a few simple connections: Independent Valuations: Both v 1 and v 2 are uniform in [0,100]. Good: Second price auction. Better: Second price auction with reserve price 50.

Anti - Correlation: v 1 is uniform in [0,100]. v 2 = v 1. Optimal: P = The maximum of (w,100-w) where w is the lower bid. Correlation: v 1 is uniform in [0,100]. v 2 = 2v 1. Bob1 is always rejected. Optimal: P = twice the lower bid.

The 1 – lookahead auction computes, based on declarations from the non-highest bidders, a price p 1: That maximizes it’s revenue from agent1 (according to ). If than agent1 wins, and pays p 1. Otherwise, nobody wins.

Theorem : the 1-lookahead auction is a 2-approximation. It satisfies IR and IC, therefore a valid auction. It’s a 2-approximation auction: splitting to two cases: and, and showing that : and The approximation ratio of 2 is tight. Sketch Of Proof:

Agent2’s type is fixed to 1. v 1 is determined acording to: The optimal revenue is about 2. Our auction generates a revenue of about 1.

When we have a polytime algorithm that can compute, given a price k and valuations (v 2,…,v n ), the probability: We can simply try for all possible k’s and choose the one that maximizes: If h is large, we can, for some α, try only the cases: (v 2, α·v 2, α 2 ·v 2,…,h), and we will get a α-approximation of the optimal price.

Vickrey Auction With Reserved Price: Let. It is the following the auction: If v 1 < r, all agents are rejected. Otherwise, agent1 wins and pays max(v 2,r).

Their exists a price r, such that the Vickrey auction with reserved price r is a 2log(h) approximation. Proof: Given a distribution d, is the expectation of v 1. Look at intervals [2 i,2 i+1 ). (log(h) such intervals). I i is the interval that contributes most to. Take r = 2 i. The revenue:

Let be the conditional distribution The K-lookahead auction is the optimal auction on agents (1,…,k) according to. Obviously, at least a 2 – approximation. The approximation ratio is tight!

Three agents, k = 2. Agent3’s type is always 1. Agent2’s type is uniformly drawn from where The probability of the type of agent1 is determined by agent2’s type. If,then with probability, and with probability. Our auction’s revenue is around. A better auction : Asks agent1 for. If, sells to agent3 for the price 1. Revenue – around 2.

Theorem: If (v 1,…,v n ) are independent, the k-lookahead auction is a -approximation. Sketch Of Proof: Fix the (n-k) lowest valuations (agents k+1,…,n). A opt is the optimal auction, R is our revenue, R opt the optimal revenue. the optimal revenue from agents (k+1,…,n). For, m j is the contribution of agent j to R opt. Case I: for all,.

Case II: Not all,. Let denote the agent with minimal m j : Pretend he declared v k+1, and run A opt on it. If any of the (n-k) won, sell to agent for v k+1. Now,. Because the distributions are independent, the distributions of the other agents don’t change.

We showed a simple 2-approximation. (1 – lookahead auction). We showed an improvement of that auction – to improve the approximation ratio to, but only under the assumption that the valuations are independent. It can be computed in polytime if there are polytime algorithms computing the distribution Φ.

Same techniques can be used to show bounds for weakly connected valuations. Finding an auction which does better than 2-approximation on general distributions (or proving it’s impossible).